A unicorn bought ten pieces of candy and paid $16. Large candies cost $4 each, medium-sized candies cost $2 a piece, and small candies come at $1 each. How many large candies did the unicorn buy?

Let be [tex]x[/tex], [tex]y[/tex] and [tex]z[/tex] the quantities of large, medium-sized and small candies bought by the unicorn., which represent non-negative integers. After a quick reading, we build the following system of linear equations:

[tex]x + y + z = 10[/tex] (1)

[tex]4\cdot x +2\cdot y +z = 16[/tex] (2)

We notice that the number of variables is greater than the number of linear equations. Thus, the system have several solutions. By (1):

[tex]z = 10 - x - y[/tex]

(1) in (2):

[tex]4\cdot x + 2\cdot y + 10 - x - y = 16[/tex]

[tex]3\cdot x + y = 6[/tex] (3)

After some inspections, we find three possible solutions:

[tex]x = 0[/tex], [tex]y = 6[/tex], [tex]z = 4[/tex]
[tex]x = 1[/tex], [tex]y = 3[/tex], [tex]z = 6[/tex]
[tex]x = 2[/tex], [tex]y = 0[/tex], [tex]z = 8[/tex]

There are three possible solutions:

6 medium-sized candies and 4 small candies.
1 large candy, 3 medium-sized candies and 6 small candies.
2 large candies and 8 small candies.

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